2003 Steve Marschner 7 Light detection discrete approx. Cone Responses S,M,L cones have broadband spectral sensitivity This sum is very clearly a dot

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1 2003 Steve Marschner Color science as linear algebra Last time: historical the actual experiments that lead to understanding color strictly based on visual observations Color Science CONTD. concrete but roundabout This time: more modern, simpler, but more abstract CS 47 Lecture 38 based on a model of how color vision works abstract but direct based more heavily on math 2003 Steve Marschner 2 A simple light detector A simple light detector Produces a scalar value (a number) when photons land on it this value depends strictly on the number of photons detected each photon has a probability of being detected that depends on the wavelength there is no way to tell the difference between signals caused by light of different wavelengths: there is just a number This model works for many detectors: based on semiconductors (such as in a digital camera) based on visual photopigments (such as in human eyes) 2003 Steve Marschner Steve Marschner 4 Light detection math Light detection math Generally work with power rather than photon count photons have power inversely proportional to wavelength so blue power is higher relative to red, but blue sensitivity is lower relative to red does not change reasoning, math, or answer for spectrum: spectral power distribution (SPD) s(λ) for detector: spectral sensitivity or spectral response r(λ) If we think of s and r as vectors, this operation is a dot product (aka inner product) in fact, the computation is done exactly this way, using sampled representations of the spectra. let λ be regularly spaced sample points λ apart; then: i this sum is very clearly a dot product 2003 Steve Marschner Steve Marschner 6

2 2003 Steve Marschner 7 Light detection discrete approx. Cone Responses S,M,L cones have broadband spectral sensitivity This sum is very clearly a dot product we won t worry too much about the λ I ll use the convention that response functions are row vectors and spectra are column vectors; in this case: S,M,L neural response is integrated w.r.t. λ we ll call the response functions r, r, r s m l results in a trichromatic visual system 2003 Steve Marschner 8 Pseudo-geometric interpretation Pseudo-geometric interpretation A dot product is a projection We are projecting a high dimensional vector (a spectrum) onto three vectors differences that a perpendicular to all 3 vectors are not detectable For intuition, we can imagine a 3D analog 3D stands in for high-d vectors 2D stands in for 3D Then vision is just projection onto a plane The information available to the visual system about a spectrum is three values this amounts to a loss of information analogous to projection on a plane Two spectra that produce the same response are metamers of one another 2003 Steve Marschner Steve Marschner 0 Color reproduction Color reproduction Say we have a spectrum s we want to match on an RGB monitor match means it looks the same any spectrum that projects to the same point in the visual color space is a good reproduction We want to compute the combination of r, g, b that will project to the same visual response as s. So, we want to find a spectrum that the monitor can produce that matches s that is, we want to display a metamer of s on the screen 2003 Steve Marschner 2003 Steve Marschner 2

3 2003 Steve Marschner 3 Donald P. Greenberg - Cornell Program of Computer Graphics Color reproduction as linear algebra Color reproduction as linear algebra The projection onto the three response functions can be written in matrix form: The spectrum that is produced by the monitor for the color signals R, G, and B is: Again the discrete form can be written as a matrix: 2003 Steve Marschner 4 Color reproduction as linear algebra Color matching functions Goal of reproduction: visual response to s and s a is the same: The rows of the color matching matrix They tell you how much of each primary is needed to match each spectral color Substituting in the expression for s a, They can be measured directly when they correspond to real primaries color matching matrix for RGB recall experiment described a lecture ago therefore they can be more preciesly known and are used as the basis for color standards nm 700 nm 546 nm 436 nm observer [Greenberg / Ferwerda] 2003 Steve Marschner Steve Marschner 6 Response Matching Functions Chromaticity R = 650 nm G = 530 nm B = 460 nm [Hunt 995] A way to reduce 3D to 2D by throwing out overall brightness. do this by normalizing the three color values so that they are independent of scale: 2003 Steve Marschner Steve Marschner 8

4 2003 Steve Marschner 9 Chromaticity diagram for RGB Defining standard matching fns. Projection onto a R = 650 nm All colors can be represented accurately in any plane in color space G = 530 nm reasonably RGB space Range of possible B = 460 nm But, for any real primaries we will have to use colors is the convex hull of the spectral [Hunt 995] negative coefficients sometimes awkward! Solution: define hypothetical primaries X, Y, Z based locus solely on their color matching functions that s the curve make sure the cmfs are always positive defined by the set of all spectral colors this results in primaries that are not real sources (i.e. they would need to have negative output at some λs) Linearly related to spectral space Only colors lying within the color triangle RGB can be achieved Steve Marschner 20 Response Matching Functions Chromaticity diag. for a different RGB 3 g R = 700 nm G = 546. nm B = nm [Hunt 995] R = 700 nm G = 546. nm B = nm 2 Only colors lying within the color triangle rgb can be achieved. [Greenberg / Ferwerda] r+g+b = r -2 - b 2003 Steve Marschner Steve Marschner 22 Color Triangle for RGB and XYZ CIEXYZ Assumptions 3 Y G. One coordinate should represent the luminance. 2 [Greenberg / Ferwerda] 2. The line between X and Y should be nearly coincident with the spectral locus for colors in the 550nm to 700nm (green-red) range. 3. The spectral locus of all realizable colors should lie in the all-positive XYZ quadrant. For realizable colors, all XYZ values and all color Z R matching functions are positive B X 2003 Steve Marschner Steve Marschner 24

5 2003 Steve Marschner 25 XYZ Color Matching Functions XYZ Color Space All positive values Equal area y is the luminous efficiency function 2003 Steve Marschner 26 Describing Color in XYZ Chromaticity Diagram Luminance Y Chromaticity X x = X + Y + Z Y y = X + Y + Z Z z = X + Y + Z ( x + y + z = ) 2003 Steve Marschner Steve Marschner 28 Chromaticity Diagram Color Gamuts Monitors/printers can t produce all visible colors Reproduction is limited to the domain within the triangle defined by the three phosphor/inks. [sources unknown] 2003 Steve Marschner Steve Marschner 30

6 2003 Steve Marschner 3 Changing color coordinates Because of linearity, it is easy to compute the XYZ values for an RGB color, given the XYZ values of the three primaries: This is essentially a change of basis, like the ones we saw in geometric transformations.